Integrand size = 24, antiderivative size = 201 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac {8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {(5 b B d+6 A b e-11 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {(4 b (5 b B d+6 A b e-11 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {\left (8 b^2 (5 b B d+6 A b e-11 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{693 e (b d-a e)^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 (a+b x)^{5/2} \left (315 B d e^2 (a+b x)^3-315 A e^3 (a+b x)^3-770 b B d e (a+b x)^2 (d+e x)+1155 A b e^2 (a+b x)^2 (d+e x)-385 a B e^2 (a+b x)^2 (d+e x)+495 b^2 B d (a+b x) (d+e x)^2-1485 A b^2 e (a+b x) (d+e x)^2+990 a b B e (a+b x) (d+e x)^2+693 A b^3 (d+e x)^3-693 a b^2 B (d+e x)^3\right )}{3465 (b d-a e)^4 (d+e x)^{11/2}} \]
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Time = 1.08 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.60
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+88 B a \,b^{2} e^{3} x^{3}-40 B \,b^{3} d \,e^{2} x^{3}+120 A a \,b^{2} e^{3} x^{2}-264 A \,b^{3} d \,e^{2} x^{2}-220 B \,a^{2} b \,e^{3} x^{2}+584 B a \,b^{2} d \,e^{2} x^{2}-220 B \,b^{3} d^{2} e \,x^{2}-210 A \,a^{2} b \,e^{3} x +660 A a \,b^{2} d \,e^{2} x -594 A \,b^{3} d^{2} e x +385 B \,a^{3} e^{3} x -1385 B \,a^{2} b d \,e^{2} x +1639 B a \,b^{2} d^{2} e x -495 b^{3} B \,d^{3} x +315 a^{3} A \,e^{3}-1155 A \,a^{2} b d \,e^{2}+1485 A a \,b^{2} d^{2} e -693 A \,b^{3} d^{3}+70 B \,a^{3} d \,e^{2}-220 B \,a^{2} b \,d^{2} e +198 B a \,b^{2} d^{3}\right )}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
default | \(-\frac {2 \left (-48 A \,b^{4} e^{3} x^{4}+88 B a \,b^{3} e^{3} x^{4}-40 B \,b^{4} d \,e^{2} x^{4}+72 A a \,b^{3} e^{3} x^{3}-264 A \,b^{4} d \,e^{2} x^{3}-132 B \,a^{2} b^{2} e^{3} x^{3}+544 B a \,b^{3} d \,e^{2} x^{3}-220 B \,b^{4} d^{2} e \,x^{3}-90 A \,a^{2} b^{2} e^{3} x^{2}+396 A a \,b^{3} d \,e^{2} x^{2}-594 A \,b^{4} d^{2} e \,x^{2}+165 B \,a^{3} b \,e^{3} x^{2}-801 B \,a^{2} b^{2} d \,e^{2} x^{2}+1419 B a \,b^{3} d^{2} e \,x^{2}-495 B \,b^{4} d^{3} x^{2}+105 A \,a^{3} b \,e^{3} x -495 A \,a^{2} b^{2} d \,e^{2} x +891 A a \,b^{3} d^{2} e x -693 A \,b^{4} d^{3} x +385 B \,a^{4} e^{3} x -1315 B \,a^{3} b d \,e^{2} x +1419 B \,a^{2} b^{2} d^{2} e x -297 B a \,b^{3} d^{3} x +315 A \,a^{4} e^{3}-1155 A \,a^{3} b d \,e^{2}+1485 A \,a^{2} b^{2} d^{2} e -693 A a \,b^{3} d^{3}+70 B \,a^{4} d \,e^{2}-220 B \,a^{3} b \,d^{2} e +198 B \,a^{2} b^{2} d^{3}\right ) \left (b x +a \right )^{\frac {3}{2}}}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{4}}\) | \(401\) |
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Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (177) = 354\).
Time = 62.91 (sec) , antiderivative size = 880, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {2 \, {\left (315 \, A a^{5} e^{3} - 8 \, {\left (5 \, B b^{5} d e^{2} - {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} e^{3}\right )} x^{5} - 4 \, {\left (55 \, B b^{5} d^{2} e - 6 \, {\left (21 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} e^{3}\right )} x^{4} + 99 \, {\left (2 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} d^{3} - 55 \, {\left (4 \, B a^{4} b - 27 \, A a^{3} b^{2}\right )} d^{2} e + 35 \, {\left (2 \, B a^{5} - 33 \, A a^{4} b\right )} d e^{2} - {\left (495 \, B b^{5} d^{3} - 11 \, {\left (109 \, B a b^{4} - 54 \, A b^{5}\right )} d^{2} e + {\left (257 \, B a^{2} b^{3} - 132 \, A a b^{4}\right )} d e^{2} - 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} e^{3}\right )} x^{3} - {\left (99 \, {\left (8 \, B a b^{4} + 7 \, A b^{5}\right )} d^{3} - 33 \, {\left (86 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} e + {\left (2116 \, B a^{3} b^{2} + 99 \, A a^{2} b^{3}\right )} d e^{2} - 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{2} - {\left (99 \, {\left (B a^{2} b^{3} + 14 \, A a b^{4}\right )} d^{3} - 11 \, {\left (109 \, B a^{3} b^{2} + 216 \, A a^{2} b^{3}\right )} d^{2} e + 15 \, {\left (83 \, B a^{4} b + 110 \, A a^{3} b^{2}\right )} d e^{2} - 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3465 \, {\left (b^{4} d^{10} - 4 \, a b^{3} d^{9} e + 6 \, a^{2} b^{2} d^{8} e^{2} - 4 \, a^{3} b d^{7} e^{3} + a^{4} d^{6} e^{4} + {\left (b^{4} d^{4} e^{6} - 4 \, a b^{3} d^{3} e^{7} + 6 \, a^{2} b^{2} d^{2} e^{8} - 4 \, a^{3} b d e^{9} + a^{4} e^{10}\right )} x^{6} + 6 \, {\left (b^{4} d^{5} e^{5} - 4 \, a b^{3} d^{4} e^{6} + 6 \, a^{2} b^{2} d^{3} e^{7} - 4 \, a^{3} b d^{2} e^{8} + a^{4} d e^{9}\right )} x^{5} + 15 \, {\left (b^{4} d^{6} e^{4} - 4 \, a b^{3} d^{5} e^{5} + 6 \, a^{2} b^{2} d^{4} e^{6} - 4 \, a^{3} b d^{3} e^{7} + a^{4} d^{2} e^{8}\right )} x^{4} + 20 \, {\left (b^{4} d^{7} e^{3} - 4 \, a b^{3} d^{6} e^{4} + 6 \, a^{2} b^{2} d^{5} e^{5} - 4 \, a^{3} b d^{4} e^{6} + a^{4} d^{3} e^{7}\right )} x^{3} + 15 \, {\left (b^{4} d^{8} e^{2} - 4 \, a b^{3} d^{7} e^{3} + 6 \, a^{2} b^{2} d^{6} e^{4} - 4 \, a^{3} b d^{5} e^{5} + a^{4} d^{4} e^{6}\right )} x^{2} + 6 \, {\left (b^{4} d^{9} e - 4 \, a b^{3} d^{8} e^{2} + 6 \, a^{2} b^{2} d^{7} e^{3} - 4 \, a^{3} b d^{6} e^{4} + a^{4} d^{5} e^{5}\right )} x\right )}} \]
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\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {13}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (177) = 354\).
Time = 0.68 (sec) , antiderivative size = 815, normalized size of antiderivative = 4.05 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{13} d^{2} e^{7} {\left | b \right |} - 16 \, B a b^{12} d e^{8} {\left | b \right |} + 6 \, A b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{2} b^{11} e^{9} {\left | b \right |} - 6 \, A a b^{12} e^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (5 \, B b^{14} d^{3} e^{6} {\left | b \right |} - 21 \, B a b^{13} d^{2} e^{7} {\left | b \right |} + 6 \, A b^{14} d^{2} e^{7} {\left | b \right |} + 27 \, B a^{2} b^{12} d e^{8} {\left | b \right |} - 12 \, A a b^{13} d e^{8} {\left | b \right |} - 11 \, B a^{3} b^{11} e^{9} {\left | b \right |} + 6 \, A a^{2} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} + \frac {99 \, {\left (5 \, B b^{15} d^{4} e^{5} {\left | b \right |} - 26 \, B a b^{14} d^{3} e^{6} {\left | b \right |} + 6 \, A b^{15} d^{3} e^{6} {\left | b \right |} + 48 \, B a^{2} b^{13} d^{2} e^{7} {\left | b \right |} - 18 \, A a b^{14} d^{2} e^{7} {\left | b \right |} - 38 \, B a^{3} b^{12} d e^{8} {\left | b \right |} + 18 \, A a^{2} b^{13} d e^{8} {\left | b \right |} + 11 \, B a^{4} b^{11} e^{9} {\left | b \right |} - 6 \, A a^{3} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )} - \frac {693 \, {\left (B a b^{15} d^{4} e^{5} {\left | b \right |} - A b^{16} d^{4} e^{5} {\left | b \right |} - 4 \, B a^{2} b^{14} d^{3} e^{6} {\left | b \right |} + 4 \, A a b^{15} d^{3} e^{6} {\left | b \right |} + 6 \, B a^{3} b^{13} d^{2} e^{7} {\left | b \right |} - 6 \, A a^{2} b^{14} d^{2} e^{7} {\left | b \right |} - 4 \, B a^{4} b^{12} d e^{8} {\left | b \right |} + 4 \, A a^{3} b^{13} d e^{8} {\left | b \right |} + B a^{5} b^{11} e^{9} {\left | b \right |} - A a^{4} b^{12} e^{9} {\left | b \right |}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{3465 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \]
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Time = 3.07 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (140\,B\,a^5\,d\,e^2+630\,A\,a^5\,e^3-440\,B\,a^4\,b\,d^2\,e-2310\,A\,a^4\,b\,d\,e^2+396\,B\,a^3\,b^2\,d^3+2970\,A\,a^3\,b^2\,d^2\,e-1386\,A\,a^2\,b^3\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {x\,\sqrt {a+b\,x}\,\left (770\,B\,a^5\,e^3-2490\,B\,a^4\,b\,d\,e^2+840\,A\,a^4\,b\,e^3+2398\,B\,a^3\,b^2\,d^2\,e-3300\,A\,a^3\,b^2\,d\,e^2-198\,B\,a^2\,b^3\,d^3+4752\,A\,a^2\,b^3\,d^2\,e-2772\,A\,a\,b^4\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {x^2\,\sqrt {a+b\,x}\,\left (-1100\,B\,a^4\,b\,e^3+4232\,B\,a^3\,b^2\,d\,e^2-30\,A\,a^3\,b^2\,e^3-5676\,B\,a^2\,b^3\,d^2\,e+198\,A\,a^2\,b^3\,d\,e^2+1584\,B\,a\,b^4\,d^3-594\,A\,a\,b^4\,d^2\,e+1386\,A\,b^5\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-22\,a\,b\,d\,e+99\,b^2\,d^2\right )\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \]
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